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Theorem c1liplem1 19343
Description: Lemma for c1lip1 19344. (Contributed by Stefan O'Rear, 15-Nov-2014.)
Hypotheses
Ref Expression
c1liplem1.a  |-  ( ph  ->  A  e.  RR )
c1liplem1.b  |-  ( ph  ->  B  e.  RR )
c1liplem1.le  |-  ( ph  ->  A  <_  B )
c1liplem1.f  |-  ( ph  ->  F  e.  ( CC 
^pm  RR ) )
c1liplem1.dv  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
c1liplem1.cn  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
c1liplem1.k  |-  K  =  sup ( ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ,  RR ,  <  )
Assertion
Ref Expression
c1liplem1  |-  ( ph  ->  ( K  e.  RR  /\ 
A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  <_  ( K  x.  ( abs `  (
y  -  x ) ) ) ) ) )
Distinct variable groups:    ph, x, y   
x, A, y    x, B, y    x, F, y
Allowed substitution hints:    K( x, y)

Proof of Theorem c1liplem1
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 c1liplem1.k . . 3  |-  K  =  sup ( ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ,  RR ,  <  )
2 imassrn 5025 . . . . . 6  |-  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  C_  ran  abs
3 absf 11821 . . . . . . 7  |-  abs : CC
--> RR
4 frn 5395 . . . . . . 7  |-  ( abs
: CC --> RR  ->  ran 
abs  C_  RR )
53, 4ax-mp 8 . . . . . 6  |-  ran  abs  C_  RR
62, 5sstri 3188 . . . . 5  |-  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  C_  RR
76a1i 10 . . . 4  |-  ( ph  ->  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  C_  RR )
8 dvf 19257 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
9 ffun 5391 . . . . . . . 8  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
108, 9ax-mp 8 . . . . . . 7  |-  Fun  ( RR  _D  F )
1110a1i 10 . . . . . 6  |-  ( ph  ->  Fun  ( RR  _D  F ) )
12 c1liplem1.dv . . . . . . . 8  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
13 cncff 18397 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( ( RR 
_D  F )  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
14 fdm 5393 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) : ( A [,] B
) --> RR  ->  dom  ( ( RR  _D  F )  |`  ( A [,] B ) )  =  ( A [,] B ) )
1512, 13, 143syl 18 . . . . . . 7  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( A [,] B ) )  =  ( A [,] B ) )
16 ssdmres 4977 . . . . . . 7  |-  ( ( A [,] B ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( A [,] B
) )  =  ( A [,] B ) )
1715, 16sylibr 203 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  dom  ( RR 
_D  F ) )
18 c1liplem1.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
1918rexrd 8881 . . . . . . 7  |-  ( ph  ->  A  e.  RR* )
20 c1liplem1.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
2120rexrd 8881 . . . . . . 7  |-  ( ph  ->  B  e.  RR* )
22 c1liplem1.le . . . . . . 7  |-  ( ph  ->  A  <_  B )
23 lbicc2 10752 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2419, 21, 22, 23syl3anc 1182 . . . . . 6  |-  ( ph  ->  A  e.  ( A [,] B ) )
25 funfvima2 5754 . . . . . . 7  |-  ( ( Fun  ( RR  _D  F )  /\  ( A [,] B )  C_  dom  ( RR  _D  F
) )  ->  ( A  e.  ( A [,] B )  ->  (
( RR  _D  F
) `  A )  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ) )
2625imp 418 . . . . . 6  |-  ( ( ( Fun  ( RR 
_D  F )  /\  ( A [,] B ) 
C_  dom  ( RR  _D  F ) )  /\  A  e.  ( A [,] B ) )  -> 
( ( RR  _D  F ) `  A
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
2711, 17, 24, 26syl21anc 1181 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  A
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
28 ffun 5391 . . . . . . 7  |-  ( abs
: CC --> RR  ->  Fun 
abs )
293, 28ax-mp 8 . . . . . 6  |-  Fun  abs
30 imassrn 5025 . . . . . . . 8  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  ran  ( RR  _D  F
)
31 frn 5395 . . . . . . . . 9  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  ran  ( RR  _D  F )  C_  CC )
328, 31ax-mp 8 . . . . . . . 8  |-  ran  ( RR  _D  F )  C_  CC
3330, 32sstri 3188 . . . . . . 7  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  CC
343fdmi 5394 . . . . . . 7  |-  dom  abs  =  CC
3533, 34sseqtr4i 3211 . . . . . 6  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  dom  abs
36 funfvima2 5754 . . . . . 6  |-  ( ( Fun  abs  /\  (
( RR  _D  F
) " ( A [,] B ) ) 
C_  dom  abs )  ->  ( ( ( RR 
_D  F ) `  A )  e.  ( ( RR  _D  F
) " ( A [,] B ) )  ->  ( abs `  (
( RR  _D  F
) `  A )
)  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ) )
3729, 35, 36mp2an 653 . . . . 5  |-  ( ( ( RR  _D  F
) `  A )  e.  ( ( RR  _D  F ) " ( A [,] B ) )  ->  ( abs `  (
( RR  _D  F
) `  A )
)  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) )
38 ne0i 3461 . . . . 5  |-  ( ( abs `  ( ( RR  _D  F ) `
 A ) )  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  =/=  (/) )
3927, 37, 383syl 18 . . . 4  |-  ( ph  ->  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  =/=  (/) )
40 ax-resscn 8794 . . . . . . . 8  |-  RR  C_  CC
41 ssid 3197 . . . . . . . 8  |-  CC  C_  CC
42 cncfss 18403 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A [,] B
) -cn-> RR )  C_  (
( A [,] B
) -cn-> CC ) )
4340, 41, 42mp2an 653 . . . . . . 7  |-  ( ( A [,] B )
-cn-> RR )  C_  (
( A [,] B
) -cn-> CC )
4443, 12sseldi 3178 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
45 cniccbdd 18821 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  (
( RR  _D  F
)  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )  ->  E. a  e.  RR  A. x  e.  ( A [,] B
) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a )
4618, 20, 44, 45syl3anc 1182 . . . . 5  |-  ( ph  ->  E. a  e.  RR  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) `  x ) )  <_ 
a )
47 fvelima 5574 . . . . . . . . . 10  |-  ( ( Fun  abs  /\  b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) )  ->  E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b )
4829, 47mpan 651 . . . . . . . . 9  |-  ( b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  ->  E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b )
49 fvelima 5574 . . . . . . . . . . . . . 14  |-  ( ( Fun  ( RR  _D  F )  /\  y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) )  ->  E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y )
5010, 49mpan 651 . . . . . . . . . . . . 13  |-  ( y  e.  ( ( RR 
_D  F ) "
( A [,] B
) )  ->  E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y )
51 fvres 5542 . . . . . . . . . . . . . . . . . . 19  |-  ( b  e.  ( A [,] B )  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 b )  =  ( ( RR  _D  F ) `  b
) )
5251adantl 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  /\  b  e.  ( A [,] B
) )  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 b )  =  ( ( RR  _D  F ) `  b
) )
5352fveq2d 5529 . . . . . . . . . . . . . . . . 17  |-  ( ( A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  /\  b  e.  ( A [,] B
) )  ->  ( abs `  ( ( ( RR  _D  F )  |`  ( A [,] B
) ) `  b
) )  =  ( abs `  ( ( RR  _D  F ) `
 b ) ) )
54 fveq2 5525 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  b  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x )  =  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  b ) )
5554fveq2d 5529 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  b  ->  ( abs `  ( ( ( RR  _D  F )  |`  ( A [,] B
) ) `  x
) )  =  ( abs `  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) `  b ) ) )
5655breq1d 4033 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  b  ->  (
( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  <->  ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  b ) )  <_  a )
)
5756rspccva 2883 . . . . . . . . . . . . . . . . 17  |-  ( ( A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  /\  b  e.  ( A [,] B
) )  ->  ( abs `  ( ( ( RR  _D  F )  |`  ( A [,] B
) ) `  b
) )  <_  a
)
5853, 57eqbrtrrd 4045 . . . . . . . . . . . . . . . 16  |-  ( ( A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  /\  b  e.  ( A [,] B
) )  ->  ( abs `  ( ( RR 
_D  F ) `  b ) )  <_ 
a )
5958adantll 694 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a )  /\  b  e.  ( A [,] B ) )  -> 
( abs `  (
( RR  _D  F
) `  b )
)  <_  a )
60 fveq2 5525 . . . . . . . . . . . . . . . 16  |-  ( ( ( RR  _D  F
) `  b )  =  y  ->  ( abs `  ( ( RR  _D  F ) `  b
) )  =  ( abs `  y ) )
6160breq1d 4033 . . . . . . . . . . . . . . 15  |-  ( ( ( RR  _D  F
) `  b )  =  y  ->  ( ( abs `  ( ( RR  _D  F ) `
 b ) )  <_  a  <->  ( abs `  y )  <_  a
) )
6259, 61syl5ibcom 211 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a )  /\  b  e.  ( A [,] B ) )  -> 
( ( ( RR 
_D  F ) `  b )  =  y  ->  ( abs `  y
)  <_  a )
)
6362rexlimdva 2667 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a )  ->  ( E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y  ->  ( abs `  y
)  <_  a )
)
6450, 63syl5 28 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a )  ->  ( y  e.  ( ( RR  _D  F
) " ( A [,] B ) )  ->  ( abs `  y
)  <_  a )
)
6564imp 418 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a )  /\  y  e.  ( ( RR  _D  F ) "
( A [,] B
) ) )  -> 
( abs `  y
)  <_  a )
66 breq1 4026 . . . . . . . . . . 11  |-  ( ( abs `  y )  =  b  ->  (
( abs `  y
)  <_  a  <->  b  <_  a ) )
6765, 66syl5ibcom 211 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a )  /\  y  e.  ( ( RR  _D  F ) "
( A [,] B
) ) )  -> 
( ( abs `  y
)  =  b  -> 
b  <_  a )
)
6867rexlimdva 2667 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a )  ->  ( E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b  -> 
b  <_  a )
)
6948, 68syl5 28 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a )  ->  ( b  e.  ( abs " ( ( RR  _D  F )
" ( A [,] B ) ) )  ->  b  <_  a
) )
7069ralrimiv 2625 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a )  ->  A. b  e.  ( abs " ( ( RR  _D  F )
" ( A [,] B ) ) ) b  <_  a )
7170ex 423 . . . . . 6  |-  ( (
ph  /\  a  e.  RR )  ->  ( A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a  ->  A. b  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) b  <_  a ) )
7271reximdva 2655 . . . . 5  |-  ( ph  ->  ( E. a  e.  RR  A. x  e.  ( A [,] B
) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  ->  E. a  e.  RR  A. b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) b  <_  a
) )
7346, 72mpd 14 . . . 4  |-  ( ph  ->  E. a  e.  RR  A. b  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) b  <_  a )
74 suprcl 9714 . . . 4  |-  ( ( ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  C_  RR  /\  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  =/=  (/)  /\  E. a  e.  RR  A. b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) b  <_  a
)  ->  sup (
( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) ,  RR ,  <  )  e.  RR )
757, 39, 73, 74syl3anc 1182 . . 3  |-  ( ph  ->  sup ( ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ,  RR ,  <  )  e.  RR )
761, 75syl5eqel 2367 . 2  |-  ( ph  ->  K  e.  RR )
77 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( A [,] B ) )
78 fvres 5542 . . . . . . . . . . 11  |-  ( y  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 y )  =  ( F `  y
) )
7977, 78syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  =  ( F `  y ) )
80 c1liplem1.cn . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
81 cncff 18397 . . . . . . . . . . . . . 14  |-  ( ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( F  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
8280, 81syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
8382ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( A [,] B
) ) : ( A [,] B ) --> RR )
8483, 77ffvelrnd 5666 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  e.  RR )
8584recnd 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  e.  CC )
8679, 85eqeltrrd 2358 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  e.  CC )
87 simplrl 736 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( A [,] B ) )
88 fvres 5542 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 x )  =  ( F `  x
) )
8987, 88syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  =  ( F `  x ) )
9083, 87ffvelrnd 5666 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  e.  RR )
9190recnd 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  e.  CC )
9289, 91eqeltrrd 2358 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  e.  CC )
9386, 92subcld 9157 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  e.  CC )
94 iccssre 10731 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
9518, 20, 94syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( A [,] B
)  C_  RR )
9695ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  RR )
9796, 77sseldd 3181 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR )
9896, 87sseldd 3181 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR )
9997, 98resubcld 9211 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR )
10099recnd 8861 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  CC )
101 simpr 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <  y )
102 difrp 10387 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  <->  ( y  -  x )  e.  RR+ ) )
10398, 97, 102syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x  <  y  <->  ( y  -  x )  e.  RR+ ) )
104101, 103mpbid 201 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR+ )
105104rpne0d 10395 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  =/=  0
)
10693, 100, 105absdivd 11937 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  =  ( ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  /  ( abs `  (
y  -  x ) ) ) )
1076a1i 10 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  C_  RR )
10839ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  =/=  (/) )
10973ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  E. a  e.  RR  A. b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) b  <_  a
)
11029a1i 10 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  Fun  abs )
11193, 100, 105divcld 9536 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  CC )
112111, 34syl6eleqr 2374 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e. 
dom  abs )
11398rexrd 8881 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR* )
11497rexrd 8881 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR* )
11598, 97, 101ltled 8967 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <_  y )
116 ubicc2 10753 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  y  e.  ( x [,] y
) )
117113, 114, 115, 116syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( x [,] y
) )
118 fvres 5542 . . . . . . . . . . . . . 14  |-  ( y  e.  ( x [,] y )  ->  (
( F  |`  (
x [,] y ) ) `  y )  =  ( F `  y ) )
119117, 118syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( x [,] y ) ) `  y )  =  ( F `  y ) )
120 lbicc2 10752 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  x  e.  ( x [,] y
) )
121113, 114, 115, 120syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( x [,] y
) )
122 fvres 5542 . . . . . . . . . . . . . 14  |-  ( x  e.  ( x [,] y )  ->  (
( F  |`  (
x [,] y ) ) `  x )  =  ( F `  x ) )
123121, 122syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( x [,] y ) ) `  x )  =  ( F `  x ) )
124119, 123oveq12d 5876 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F  |`  (
x [,] y ) ) `  y )  -  ( ( F  |`  ( x [,] y
) ) `  x
) )  =  ( ( F `  y
)  -  ( F `
 x ) ) )
125124oveq1d 5873 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  =  ( ( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) ) )
126 iccss2 10720 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) )  -> 
( x [,] y
)  C_  ( A [,] B ) )
127126ad2antlr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x [,] y )  C_  ( A [,] B ) )
128 resabs1 4984 . . . . . . . . . . . . . . 15  |-  ( ( x [,] y ) 
C_  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  =  ( F  |`  ( x [,] y ) ) )
129127, 128syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  =  ( F  |`  ( x [,] y ) ) )
13080ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> RR ) )
131 rescncf 18401 . . . . . . . . . . . . . . 15  |-  ( ( x [,] y ) 
C_  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) ) )
132127, 130, 131sylc 56 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) )
133129, 132eqeltrrd 2358 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) )
13440a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  RR  C_  CC )
135 c1liplem1.f . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F  e.  ( CC 
^pm  RR ) )
136135ad2antrr 706 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F  e.  ( CC  ^pm  RR ) )
137 cnex 8818 . . . . . . . . . . . . . . . . . . . 20  |-  CC  e.  _V
138 reex 8828 . . . . . . . . . . . . . . . . . . . 20  |-  RR  e.  _V
139137, 138elpm2 6799 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
140139simplbi 446 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  ( CC  ^pm  RR )  ->  F : dom  F --> CC )
141136, 140syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F : dom  F --> CC )
142139simprbi 450 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
143136, 142syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  F  C_  RR )
144 iccssre 10731 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x [,] y
)  C_  RR )
14598, 97, 144syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x [,] y )  C_  RR )
146 eqid 2283 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
147146tgioo2 18309 . . . . . . . . . . . . . . . . . 18  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
148146, 147dvres 19261 . . . . . . . . . . . . . . . . 17  |-  ( ( ( RR  C_  CC  /\  F : dom  F --> CC )  /\  ( dom  F  C_  RR  /\  (
x [,] y ) 
C_  RR ) )  ->  ( RR  _D  ( F  |`  ( x [,] y ) ) )  =  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) ) ) )
149134, 141, 143, 145, 148syl22anc 1183 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( RR  _D  ( F  |`  (
x [,] y ) ) )  =  ( ( RR  _D  F
)  |`  ( ( int `  ( topGen `  ran  (,) )
) `  ( x [,] y ) ) ) )
150 iccntr 18326 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) )  =  ( x (,) y
) )
15198, 97, 150syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  (
x [,] y ) )  =  ( x (,) y ) )
152151reseq2d 4955 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) ) )  =  ( ( RR 
_D  F )  |`  ( x (,) y
) ) )
153149, 152eqtrd 2315 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( RR  _D  ( F  |`  (
x [,] y ) ) )  =  ( ( RR  _D  F
)  |`  ( x (,) y ) ) )
154153dmeqd 4881 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( RR 
_D  ( F  |`  ( x [,] y
) ) )  =  dom  ( ( RR 
_D  F )  |`  ( x (,) y
) ) )
155 ioossicc 10735 . . . . . . . . . . . . . . . . 17  |-  ( x (,) y )  C_  ( x [,] y
)
156155, 127syl5ss 3190 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x (,) y )  C_  ( A [,] B ) )
15717ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  dom  ( RR  _D  F
) )
158156, 157sstrd 3189 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x (,) y )  C_  dom  ( RR  _D  F
) )
159 ssdmres 4977 . . . . . . . . . . . . . . 15  |-  ( ( x (,) y ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( x (,) y
) )  =  ( x (,) y ) )
160158, 159sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( ( RR  _D  F )  |`  ( x (,) y
) )  =  ( x (,) y ) )
161154, 160eqtrd 2315 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( RR 
_D  ( F  |`  ( x [,] y
) ) )  =  ( x (,) y
) )
16298, 97, 101, 133, 161mvth 19339 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  E. a  e.  ( x (,) y
) ( ( RR 
_D  ( F  |`  ( x [,] y
) ) ) `  a )  =  ( ( ( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) ) )
163153fveq1d 5527 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( RR  _D  ( F  |`  ( x [,] y
) ) ) `  a )  =  ( ( ( RR  _D  F )  |`  (
x (,) y ) ) `  a ) )
164163adantrr 697 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( RR  _D  ( F  |`  ( x [,] y ) ) ) `  a )  =  ( ( ( RR  _D  F )  |`  ( x (,) y
) ) `  a
) )
165 fvres 5542 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  ( x (,) y )  ->  (
( ( RR  _D  F )  |`  (
x (,) y ) ) `  a )  =  ( ( RR 
_D  F ) `  a ) )
166165ad2antll 709 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( ( RR 
_D  F )  |`  ( x (,) y
) ) `  a
)  =  ( ( RR  _D  F ) `
 a ) )
167164, 166eqtrd 2315 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( RR  _D  ( F  |`  ( x [,] y ) ) ) `  a )  =  ( ( RR 
_D  F ) `  a ) )
16810a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  ->  Fun  ( RR  _D  F
) )
16917ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( A [,] B
)  C_  dom  ( RR 
_D  F ) )
170156sseld 3179 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( a  e.  ( x (,) y
)  ->  a  e.  ( A [,] B ) ) )
171170impr 602 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
a  e.  ( A [,] B ) )
172 funfvima2 5754 . . . . . . . . . . . . . . . . . 18  |-  ( ( Fun  ( RR  _D  F )  /\  ( A [,] B )  C_  dom  ( RR  _D  F
) )  ->  (
a  e.  ( A [,] B )  -> 
( ( RR  _D  F ) `  a
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) ) )
173172imp 418 . . . . . . . . . . . . . . . . 17  |-  ( ( ( Fun  ( RR 
_D  F )  /\  ( A [,] B ) 
C_  dom  ( RR  _D  F ) )  /\  a  e.  ( A [,] B ) )  -> 
( ( RR  _D  F ) `  a
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
174168, 169, 171, 173syl21anc 1181 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( RR  _D  F ) `  a
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
175167, 174eqeltrd 2357 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( RR  _D  ( F  |`  ( x [,] y ) ) ) `  a )  e.  ( ( RR 
_D  F ) "
( A [,] B
) ) )
176 eleq1 2343 . . . . . . . . . . . . . . 15  |-  ( ( ( RR  _D  ( F  |`  ( x [,] y ) ) ) `
 a )  =  ( ( ( ( F  |`  ( x [,] y ) ) `  y )  -  (
( F  |`  (
x [,] y ) ) `  x ) )  /  ( y  -  x ) )  ->  ( ( ( RR  _D  ( F  |`  ( x [,] y
) ) ) `  a )  e.  ( ( RR  _D  F
) " ( A [,] B ) )  <-> 
( ( ( ( F  |`  ( x [,] y ) ) `  y )  -  (
( F  |`  (
x [,] y ) ) `  x ) )  /  ( y  -  x ) )  e.  ( ( RR 
_D  F ) "
( A [,] B
) ) ) )
177175, 176syl5ibcom 211 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( ( RR 
_D  ( F  |`  ( x [,] y
) ) ) `  a )  =  ( ( ( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  ->  (
( ( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  e.  ( ( RR  _D  F
) " ( A [,] B ) ) ) )
178177expr 598 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( a  e.  ( x (,) y
)  ->  ( (
( RR  _D  ( F  |`  ( x [,] y ) ) ) `
 a )  =  ( ( ( ( F  |`  ( x [,] y ) ) `  y )  -  (
( F  |`  (
x [,] y ) ) `  x ) )  /  ( y  -  x ) )  ->  ( ( ( ( F  |`  (
x [,] y ) ) `  y )  -  ( ( F  |`  ( x [,] y
) ) `  x
) )  /  (
y  -  x ) )  e.  ( ( RR  _D  F )
" ( A [,] B ) ) ) ) )
179178rexlimdv 2666 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( E. a  e.  ( x (,) y ) ( ( RR  _D  ( F  |`  ( x [,] y
) ) ) `  a )  =  ( ( ( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  ->  (
( ( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  e.  ( ( RR  _D  F
) " ( A [,] B ) ) ) )
180162, 179mpd 14 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  e.  ( ( RR  _D  F
) " ( A [,] B ) ) )
181125, 180eqeltrrd 2358 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  ( ( RR  _D  F ) " ( A [,] B ) ) )
182 funfvima 5753 . . . . . . . . . . 11  |-  ( ( Fun  abs  /\  (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) )  e.  dom  abs )  ->  ( ( ( ( F `  y )  -  ( F `  x ) )  / 
( y  -  x
) )  e.  ( ( RR  _D  F
) " ( A [,] B ) )  ->  ( abs `  (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) ) )  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ) )
183182imp 418 . . . . . . . . . 10  |-  ( ( ( Fun  abs  /\  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) )  e.  dom  abs )  /\  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) )  e.  ( ( RR 
_D  F ) "
( A [,] B
) ) )  -> 
( abs `  (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) ) )  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) )
184110, 112, 181, 183syl21anc 1181 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) )
185 suprub 9715 . . . . . . . . 9  |-  ( ( ( ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  C_  RR  /\  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  =/=  (/)  /\  E. a  e.  RR  A. b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) b  <_  a
)  /\  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) )  ->  ( abs `  (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) ) )  <_  sup (
( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) ,  RR ,  <  ) )
186107, 108, 109, 184, 185syl31anc 1185 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  <_  sup (
( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) ,  RR ,  <  ) )
187186, 1syl6breqr 4063 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  <_  K )
188106, 187eqbrtrrd 4045 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( abs `  ( ( F `
 y )  -  ( F `  x ) ) )  /  ( abs `  ( y  -  x ) ) )  <_  K )
18993abscld 11918 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  e.  RR )
19076ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  K  e.  RR )
191100, 105absrpcld 11930 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( y  -  x
) )  e.  RR+ )
192189, 190, 191ledivmuld 10439 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  /  ( abs `  ( y  -  x
) ) )  <_  K 
<->  ( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  <_  ( ( abs `  ( y  -  x ) )  x.  K ) ) )
193188, 192mpbid 201 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  (
( abs `  (
y  -  x ) )  x.  K ) )
194191rpcnd 10392 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( y  -  x
) )  e.  CC )
195190recnd 8861 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  K  e.  CC )
196194, 195mulcomd 8856 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( abs `  ( y  -  x ) )  x.  K )  =  ( K  x.  ( abs `  ( y  -  x
) ) ) )
197193, 196breqtrd 4047 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  ( K  x.  ( abs `  ( y  -  x
) ) ) )
198197ex 423 . . 3  |-  ( (
ph  /\  ( x  e.  ( A [,] B
)  /\  y  e.  ( A [,] B ) ) )  ->  (
x  <  y  ->  ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  <_  ( K  x.  ( abs `  ( y  -  x ) ) ) ) )
199198ralrimivva 2635 . 2  |-  ( ph  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  <_  ( K  x.  ( abs `  (
y  -  x ) ) ) ) )
20076, 199jca 518 1  |-  ( ph  ->  ( K  e.  RR  /\ 
A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  <_  ( K  x.  ( abs `  (
y  -  x ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   class class class wbr 4023   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   supcsup 7193   CCcc 8735   RRcr 8736    x. cmul 8742   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   RR+crp 10354   (,)cioo 10656   [,]cicc 10659   abscabs 11719   TopOpenctopn 13326   topGenctg 13342  ℂfldccnfld 16377   intcnt 16754   -cn->ccncf 18380    _D cdv 19213
This theorem is referenced by:  c1lip1  19344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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